About Rationally Speaking

Rationally Speaking is a blog maintained by Prof. Massimo Pigliucci, a philosopher at the City University of New York. The blog reflects the Enlightenment figure Marquis de Condorcet's idea of what a public intellectual (yes, we know, that's such a bad word) ought to be: someone who devotes himself to "the tracking down of prejudices in the hiding places where priests, the schools, the government, and all long-established institutions had gathered and protected them." You're welcome. Please notice that the contents of this blog can be reprinted under the standard Creative Commons license.

As is well known, Hume wasn’t very keen on metaphysics in general. One of the most famous quotes by him (in section 12 of the very same Enquiry) says: “If we take in our hand any volume; of divinity or school metaphysics, for instance; let us ask, Does it contain any abstract reasoning concerning quantity or number? No. Does it contain any experimental reasoning concerning matter of fact and existence? No. Commit it then to the flames: for it can contain nothing but sophistry and illusion.” Ouch.

Anyway, back to metaphysical necessity. What might it mean for something to be metaphysically necessary, or — conversely — metaphysically impossible? Not surprisingly, there is a fairly large literature about this. The (far from comprehensive, but heavy on recent entries) section on metaphysical necessity of the PhilPapers archive lists 74 papers, with some of the most recent entries having titles like “Hume’s Dictum and Natural Modality: Counterfactuals”; “Radical Non-Dispositionalism and the Permutation Problem”; “Soames’s Deflationism About Modality”; and so forth.

But we’ll proceed here by looking briefly at the basics. First of all, metaphysical necessity/impossibility as opposed to what other kinds of necessity/impossibility? Two immediately come to mind: logical and physical. It is logically necessary that I either am me or am not-me, for instance [1]; it is also logically necessary, though for different reasons, that there is no such thing as a married bachelor. It is physically necessary that objects with mass attract each other; it is also physically impossible for me to both be here in New York and simultaneously in Rome [2]. And so forth.

Let’s see what we can glean from the above examples: in both instances concerning physical possibilities, and in one instance concerning logical possibility, the idea seems to be that there are certain “laws” that govern logic or physics, and that these laws are inviolable. Now, one could be skeptical about the a priori validity of the laws of logic (like W.V.O. Quine was), and one can even think of the laws of physics as simply empirical generalizations that could, in fact, admit of exceptions or have a limited domain of application (like, for instance, Nancy Cartwright does), but I won’t go there. As far as we are concerned, both logic and physics are solid enough, so to speak, to allow us to talk about things that are either possible or impossible given the respective sets of laws.

The remaining case (the impossibility of a married bachelor), of course, hinges on issues of definitions: since a bachelor is defined as an unmarried man, there simply cannot be any such thing as a married one, on penalty of (logical-semantic) contradiction. Definitions, of course, are tautological, and tautologies are often regarded with little interest in such discussions. But this is a mistake: think about the fact that mathematics (and much of logic itself) consists precisely in the working out of the tautological implications of certain axioms or premises.

So, where were we? Well, the discussion so far hints at one promising way to look for metaphysical necessity: search for laws of metaphysics. Unfortunately, that’s not at all a straightforward quest, because it is not clear what counts as a metaphysical law, as distinct from either a physical law or a law of logic — which of course doesn’t help our predicament at all.

Perhaps we should do what I’ve done above in the cases of logic and physics: look for examples first, then see what we can learn from them.

If you follow that route, one of the most commonly advanced examples of metaphysical necessity is… the existence of God! Since that is prima facie (I love it when I get to write that!) ludicrous — or it should be at the dawn of the 21st century — we will ignore it and proceed otherwise.

What else can be done? Well, there are some more intriguing examples of alleged metaphysical necessity, for instance “whatever is water is H2O” and “whatever is elemental gold has atomic number 79.”

Let’s look more closely: these are not examples of definitional necessity, like the bachelor. True, once we discovered that the molecular structure of water is H2O we could simply define water as that substance that has that chemical structure and be done with it. But this required an empirical discovery, it wasn’t true a priori from the get go, as is the fact that there cannot be a married bachelor. The reasoning is the same for gold being the element with atomic number 79.

Could it be that these two examples can be interpreted as instantiations of the laws of logic? Hard to see how. There is nothing logically contradictory in imagining a substance with the characteristics of water that is not made of two atoms of hydrogen and one of oxygen. But wouldn’t that contradict the laws of physics, at least? Ah, here things become tricky. Surely water behaves the way it is in our universe because the laws of physics are such that if a molecule has that structure then it will behave in that way. But it is hard to say which specific law of physics would be violated if something made of H2O actually behaved differently (say, it had a different freezing point at standard pressure).

Another way to think about this is to say that we can imagine a universe where the physics is (slightly) different and where, as a consequence, H2O doesn’t behave as our H2O. Of course, if that were the case, the H2O = water equation would not be a metaphysical necessity after all, but only a physical one. That’s because metaphysicians these days seem to make sense of the notion of metaphysical necessity by saying that something is metaphysically necessary if it is true in all possible worlds.

Talk of possible worlds is tightly connected with modal logic which, not surprisingly, is a set of logics that deal with expressions such as “necessarily,” “possibly,” etc. — which philosophers call modalities. There are a bunch of modal logics, including deontic (dealing with what is morally necessary or permissible), temporal, conditional and so forth. These have given origin to what is known as possible worlds semantics, the study of logical languages that make it possible for logicians to determine whether a given modal expression is inferentially valid or not (which, after all, is the whole point of any logic).

To return to our example: is it physically or metaphysically necessary that H2O = water? For this to be an example of metaphysical necessity, the equation would have to be valid in all possible worlds. But what makes a world possible to begin with? We could, again be talking about either logical or physical possibility (the former, should be clear, being much ampler than the latter). Let’s say we are talking about physical possibility: possible worlds are those worlds that could exist while instantiating a coherent set of physical laws.

Our world, obviously, realizes one of these possibilities. Worlds that, say, were different from ours only with respect to the gravitational constant would be our possible-neighbors, the closer to us as a function of how similar their gravitational constant is to ours.

One can easily extend this concept to a multidimensional landscape of fundamental physical constants, each varying within whatever range is physically possible for them to vary (e.g., although logically the gravitational constant could take any of an infinite number of values, it is perfectly possible that only a small subset of these values would yield a physically realizable universe).

If you smelled “multiverse” you are close. Despite some people’s reservations about the scientific status of the multiverse theory (reservations with which I sympathize), it does seem to make philosophical sense to deploy it within the context of this discussion. But if you don’t like that particular take, then think of possible worlds as the set of worlds that are mathematically realizable instead. While neither of these senses is the one normally used by philosophers who are interested in possible worlds semantics, I think they do help to get an intuitive grasp on the whole idea of “possible worlds,” because they give a fairly precise answer to the obvious question: possible in what sense?

So, again, water = H2O would be a metaphysical necessity just in case it had to be true in all possible worlds, say in the entire multiverse. My hunch is that this isn’t the case. It seems that some change in one physical constant or another would yield a pocket universe (within the multiverse) where a substance had the molecular structure H2O and yet had different physical characteristics from our water.

Still, there may be things that are metaphysically necessary in all possible instantiations of the multiverse. Perhaps the inter-conversion between matter and energy? Or the existence of fields from which matter emerges (like the Higgs)?

I am going to bet that most metaphysicians won’t like my analysis of metaphysical necessity as presented above, though. True, I have arrived at the conclusion that there can be such a thing as metaphysical necessity as smaller than logical necessity but ampler than physical necessity, thus legitimizing the concept. But I have also linked said concept operationally to either the multiverse as conceived by modern physics or its mathematical equivalent. If so, then discovering metaphysical necessities becomes either a matter for physics (because it is an empirical question) or for logicians-mathematicians (because it is a logical-mathematical thing). Which means that even our newfound way of thinking about metaphysical necessity either expands into logical necessity or collapses into physical necessity.

At the least, that’s the way I see it this week. Anyone out there have examples of metaphysical necessity that would rescue the concept from the Scylla or logic and the Charybdis of physics?Postscript on the role of metaphysics

Interesting discussion so far. I wanted to add a few notes to further refine my thoughts about this issue. To begin with, I am leaning toward the conclusion that there is no such thing as metaphysical necessity. That’s in part because one cannot find metaphysical laws, and in part because I doubt there is such a thing as necessity, period. Nothing is physically or logically necessary - only possible or impossible.

True, once we establish certain constraints - for instance the laws of physics in our universe - then certain things necessarily happen. (Indeed, if you are a determinist, everything necessarily happens.) But there doesn’t seem to be a reason to think that the laws of physics themselves are necessary (multiverse and all that), so…

The same goes with logical necessity: once we pick certain axioms or premises, a number of things necessarily follow. But we could have picked different axioms or premises, so that those very same things wouldn’t follow at all.

Where, then, does that leave metaphysics? I still think it has a role to play, in the same sense that philosophy in general has a role to play. I have come to see philosophy as a type of critical inquiry that bridges logic (broadly construed) and science (and other sources of empirical knowledge), in the sense that it applies rigorous reasoning to whatever the issue at hand may be (e.g., ethics) while taking into account empirical input. This is nothing new: it is a restatement of Kan’t compromise between rationalism (the idea that one can derive a priori truths about the world) and empiricism (the idea that all truths derive from sense experience).

Similarly for metaphysics: I see it as a bridge between the Scylla of logic and the Charybdis of physics: the role of metaphysics is to make reasoned sense of what the natural sciences tell us about the world (in this I’m with people like Ladyman and Ross), as well as to elucidate how that knowledge fits with our understanding of abstract objects, such as mathematical and logical relations. But there are no laws of metaphysics, just like there are no laws of philosophy, so this endeavor is one of critically making sense of things, not of discovering or dictating how things are.

At least (again), this is what I think this week...

——

[1] For the purposes of this discussion I will assume standard classical logic. The details would be different, but the general arguments the same, if we were using other kinds of logics.[2] Non-locality does not apply to macroscopic objects of the size of a human being, for reasons that not even quantum physicists are particularly sure of.

124 comments:

Well here's a possible testing ground: is the world of Harry Potter a possible world? if not why not? So normally we assume magic (as in Hogwarts) is not possible. Is that a physical impossibility or a metaphysical impossibility?It all hinges on how one thinks of the laws of physics - but why should the laws of physics be of the kind that we experience in this particular patch of the universe?

The world of Harry Potter is not logically possible because the rules of magic are inconsistent and/or ill-defined.

Think about how problematic it is even to form a well-defined rule that something magical happens when you utter the words "Wingardium Leviosa" but not if you say instead "Wingardum Levosa"? What makes the judgement of whether you have pronounced it correctly, or loud enough, or emphatically enough? What if you happen to have a thick accent, or your voice is unusually deep or squeaky? What is it that is listening to you and making this judgement? The rules as stated just don't work.

I recommend the excellent "Harry Potter and the Methods of Rationality" by Eliezer Yudkowsky, and in particular the opening chapters, for a fun exploration of the enjoyable nonsense that is the world of Harry Potter and fictional magic in general.

I've also written about why the supernatural is logically impossible on my blog.

I am not sure about Harry Potter, but when world-building for an alternate world fantasy novel I was writing, I had a physicist check out my magic system - or rather the alternate system of physics that allows the magic in the world, and it seems it was entirely physically consistent. (Think of something like Terry Pratchett's thaums, i.e. magic particles.)Working magic in this case is analogous to playing an instrument, or causing vibrations in a solid through the use of your voice. The words mean nothing, but if you do it wrong it doesn't work. (That's the way it works in most good fantasy, i.e not in Harry Potter ;-) )

Methods of Rationality wasn't logically impossible until I had a floor of the hallway tiled in pentagons, and made the apparent number of objects in Dumbledore's office change without any being added or subtracted. Until that point, nothing happened inside the story that you couldn't easily make happen inside a sufficiently good simulated universe.

My point is not that Methods of Rationality is impossible but that Harry Potter is, at least a naive interpretation of it.

It's not so much that Harry Potter is certainly logically impossible, it's more that a lot of apparent contradictions and hand-wavy stuff would need to be clarified before it could be made workable. This is why I particularly appreciated it when you write about how crazy it is that Professor McGonagall can transform into a cat. Where do all the extra atoms go? How do they know how to rearrange themselves? How can her brain maintain the same mind when it changes size and shape so drastically? That kind of thing.

Magic in Harry Potter works by storybook logic by treating high level concepts such as people and cats as primitives, but any consistent logical universe needs to have laws which take as primitives low-level objects such as fundamental particles, and it's not easy to see how to do that with Harry Potter.

Really you just appear to be saying that Harry Potter world couldn't be the case in our physical world.

I cannot see why you could not have a consistent Harry Potter world. I think it is crazy that all the cells in my body know what sort of cells to turn into and what to do, but it is the case.

I cannot see why you could not make a consistent Harry Potter world. The incantation stuff is easy, no more difficult than the speech recognition in my kids' toy wands.

The transformation into a cat might be challenging, but not impossible. If consciousness is a computation, as you say, then it makes things easier, there just needs to be an equivalent computation in a different brain. It does not matter that McGonagall might be experiencing a sort of more intelligent version of cat consciousness. Then you would just have to worry about the transitional phase - there might be a slight gap in consciousness or there might be a calculation of transitional phases that maintain the same algorithm.

Where to the extra atoms go? Maybe it is the type of physical reality where production and annihilation can happen at the atomic level.

How do they know to rearrange themselves? Maybe it is the sort of universe where atoms can carry morphic information as our cells do and that by thinking certain thoughts McGonagall can trigger a transition from one to the other.

I am sure Ms Rowling had nothing like this in mind, but I cannot see that you could call it logically impossible.

In fact there is a universe just like it in your Mathematical Universe :)

Again, I'm not saying it is categorically impossible. I am saying that the rules as stated are too simple, too naive. The rules as presented are not workable in the state they are in and need to be refined and made more precise in order for us to figure out if they make sense. In the vague, storybook terms in which the rules of magic are described in Harry Potter, it's very hard to see if they are internally consistent, and especially if they are consistent with a universe which is from a muggle perspective identical to our own.

The answers you give to the questions I posed might all be plausible, but each one of them raises more. For example, if there is speech recognition software floating in the ether, then does it require actual sound vibrations or is it the lip movements that matter? If it's sound vibrations, then will a recording do? If not, then what is it about live speech that makes the difference?

Etc, etc, etc. We might bottom out at a sensible rigorous account at some point, but it's not clear that we will, and we may run afoul of logical inconsistency eventually.

My position is that the supernatural is impossible. If we do manage to come up with a coherent model for how magic works in Harry Potter, then in my view it is no longer supernatural. It is by definition natural because we have just come up with a universe with different physical laws which are just as rigorous and mathematical as our own.

DM wrote: "Again, I'm not saying it is categorically impossible. I am saying that the rules as stated are too simple, too naive. The rules as presented are not workable in the state they are in and need to be refined and made more precise in order for us to figure out if they make sense.

Hey George,I just saw Massimo’s tweet regarding this new blog entry. I haven’t read it yet. Nevertheless I was able to read your comment and I know enough about “life the universe and everything” to have already formed an opinion. The world of Harry Potter is not possible because it cannot be verified. The laws of physics are only laws because they can be verified and have been verified (and refined) over and over again. If a theory (or in this case a piece of popular fiction) contradicts our so called “laws of physics” then we might say that these ideas are beyond the realm of “possibility” because they go against all the observations and evidence that has ever been collected, verified and written down.

The idea that the “laws” are different in different patches of the universe is also one that cannot be verified. The history of the development of the laws of physics shows us that it’s prudent to assume that the laws are incomplete and that things will behave different under circumstance (and patches of the universe) that we know nothing about. The idea that matter behaves differently in different patches of the universe is an interesting thought, but until it can be verified it has more in common with the magic in Harry Potter than Newton’s Laws.

Your argument shows that the identity "water=H20" might not be necessary, but still "if something is water, then it has molecular structure H20" could be necessary. Something which superficially look just like water, but does not have the same molecular structure, would not be "our" water and should not be given the same name, because the term "water" never referred to a bundle of superficial manifestations but to the real entity which causes these manifestations (=H20). This is what Kripke's semantic arguments show.

So metaphysically speaking, H2O could have not been water, but water could not have not been H2O.

Another of Kripke's examples is the necessity of origin: something (a table, a person) couldn't have a different origin (its wood, one's parents) otherwise it would not be the same thing.Denying the necessity of origin is not absurd (it's not a logical or analytical truth), but it makes it very difficult to think of identity in a coherent manner: origin and identity are somehow related concepts, but their relation is not purely formal or definitional as per logical truths.

So one might view metaphysical necessities as conceptual necessities : true in virtue of the meaning of our concepts, but yet not tautological "by definition", or purely formal truth. Kant's view that metaphysical truths are synthetic a priori (true in virtue of the world, but known by intuition rather than experience) could also be of interest here.

“If we take in our hand any volume; of divinity or school metaphysics, for instance; let us ask, Does it contain any abstract reasoning concerning quantity or number? No. Does it contain any experimental reasoning concerning matter of fact and existence? No. Commit it then to the flames: for it can contain nothing but sophistry and illusion.” Ouch

How much of the Universe can One hold in his hand? Is it even measurable and does it really matter? Beyond physics there is a field where the hand of One is the Universe of all. I'll meet you there. =

If there is a substance like water in another universe but it's not H20, then I don't think it is water. If there is H20 in another universe but it behaves differently because the laws of physics are different, firstly I'm not convinced that it's composed of "genuine" hydrogen and oxygen but some alien analogues thereof, and secondly if you could convince me that it was composed of hydrogen and oxygen then I would indeed regard it as water.

Whichever way you analyse it, this question seems to me to be not metaphysical at all and it only depends on what you mean by water. I mean "H20" by water. If you mean "a watery substance" then you simply mean something else and you will get a different answer. As such, it's not a very interesting question unless your domain of interest is the semantics of the word "water". It's a linguistic rather than a metaphysical question in other words.

Personally, I reject the idea of metaphysical necessity as distinct from logical necessity. For something to be metaphysically necessary we would need a set of metaphysical laws. These laws could not be deduced from first principles because this would reduce them to logical necessity. They could not be detected by scientists because this would reduce them to physical laws. A metaphysical law would be something like "The value of the gravitational constant G may vary from [x] to [y]", but it would only be a truly metaphysical law if there was no way in principle for us to discover it.

This is one of many reasons why I think all logically possible worlds must exist.

I'm reading this just after I had a discussion about metaphysical necessity on another blog. The author claimed that it was a metaphysical necessity that the universe had an atemporal cause. The cause, of course, was god.

The discussion learned me something about metaphysics. The starting point usually is something rather uncontroversial or even bland. Water is H2O, if I burn my fingers there was a cause - I touched a hot stove - etc. etc.

Then there's a process of abstraction, a move to "generality". Suddenly a cause becomes a Cause and water becomes Water. It becomes tempting to define what a Cause - metaphysically speaking - "really" is. A Cause becomes a thing on its own, existing independently of hot stoves and burned fingers.

In the end there's a metaphysical necessity. But that arrives usually far past the moment when it became doubtful that all these Causes and Waters were describing anything except their own life in the metaphysical universe.

>Does this mean that the non-existence of a logically possible world would entail a contradiction?<

Yes, because the concept of physical existence doesn't work when discussing universes (I explain why on my blog). All that is left is mathematical Platonic existence. For a logically possible world to not exist, it would have to be mathematically inconsistent. This is contradictory.

>if all logically possible worlds exist, then what exactly is it that instantiates them?<

Nothing instantiates them or they instantiate themselves. They exist necessarily, like numbers.

Massimo, I'm out of my depth here, but isn't this a discussion of subjunctive possibility, about which I'm largely confused? I was reading (struggling with) an article on supervenience in SEP the other day and your use of water (= H2O) reminded me of it. In the article, it is said: "Finally, we will assume that whatever is metaphysically necessary is nomologically necessary, but not conversely. (One can hold that there are nomologically necessary truths that are not metaphysically necessary, even if one holds that all nomic properties play their nomic roles essentially; see Fine 2002.)." [http://plato.stanford.edu/entries/supervenience/]

This leads me to wonder whether your statement "then discovering metaphysical necessities becomes either a matter for physics (because it is an empirical question) or for logicians-mathematicians (because it is a logical-mathematical thing)" leaves space for some confusion regarding metaphysical and nomological necessities. So I guess what I'm asking is whether an empirical approach might reveal a nomological necessity that is mistaken for a metaphysical necessity, but how would one know the difference.

Now a universe in which the structure and properties of atoms (which would include the structure and properties of all sub atomic identities) is identical to those in our own world but the macro behaviour is different?

The problem with what you are saying is that it implies that there is some necessarily true statement of pure logic that rules out the existence of things that are necessary but not logically necessary.

An interesting post. I'm almost certainly missing something here, but it seems to me that the only metaphysical *necessities* are tautological ones. Physical necessities imply, to me at least, that we're no longer talking about metaphysics, but science of some sort. I realize this might hinge on your definition of the term 'metaphysics'.

I also have to agree with Disagreeable Me that if you start changing the physical laws governing H2O in another universe, that we're not talking about water anymore, and depending on the changes, we might not even be talking about hydrogen or oxygen, or conceivably even atoms and molecules.

SelfAwarePatterns wrote: " here, but it seems to me that the only metaphysical *necessities* are tautological ones. Physical necessities imply, to me at least, that we're no longer talking about metaphysics, but science of some sort."

The problem with that is that Massimo does not appear to be talking about things that are necessary, given the fact of something else, but things that are necessarily true at all possible worlds.

Science ultimately has two tools - observation and mathematics (which includes logic).

Self evident a necessary truth cannot be verified inductively and so we are basically back to mathematics.

So what we are talking about here is not so much science as the search for an Ontological Argument.

> is the world of Harry Potter a possible world? if not why not? So normally we assume magic (as in Hogwarts) is not possible. Is that a physical impossibility or a metaphysical impossibility? <

Harry’s world is certainly physically impossible, but of course that doesn’t settle the question. As long as it doesn’t include logical contradictions, however, it would be logically possible. The issue, again, is whether there is some interesting notion of possibility in between.

Patrick,

> The world of Harry Potter is not possible because it cannot be verified. <

I disagree. That seems to confuse metaphysics (is the Potter world possible?) with epistemology (can we know about it?). While I generally tend to think that one’s metaphysics better go hand in hand in one’s epistemology, the two are nonetheless distinct.

> The idea that the “laws” are different in different patches of the universe is also one that cannot be verified. <

And yet, it is been taken very seriously by some physicists, at the least those enamored of multiverses and similar concepts.

Quentin,

> Something which superficially look just like water, but does not have the same molecular structure, would not be "our" water and should not be given the same name, because the term "water" never referred to a bundle of superficial manifestations but to the real entity which causes these manifestations (=H20). This is what Kripke's semantic arguments show. <

Then I think the problem is with Kripke’s semantics (ouch!). More specifically, my hypothetical situation is one in which a substance does have the “essential” molecular characteristic of being H2O and yet, because the laws of physics are different, doesn’t have the same physical characteristics of “our” water. I care little for what we would call such substance, but the possibility raised seems to be that water = H2O is not a metaphysical necessity, as long as by water we don’t mean (tautologically) just H2O but a substance with certain physical characteristics.

> So one might view metaphysical necessities as conceptual necessities: true in virtue of the meaning of our concepts, but yet not tautological "by definition", or purely formal truth. <

Yeah, I’m sympathetic to this way of looking at it, and I don’t object to your Kripkean example concerning personal identity (which I see as distinct from the water issue above).

Even departing from a pure kripkean semantic (=assuming that something must have some physical characteristics to be water) my point was that "water implies H2O" could still be considered a metaphysical necessity, even though "H2O implies water" is not necessary...

> If there is H20 in another universe but it behaves differently because the laws of physics are different, firstly I'm not convinced that it's composed of "genuine" hydrogen and oxygen but some alien analogues thereof, and secondly if you could convince me that it was composed of hydrogen and oxygen then I would indeed regard it as water. <

Then you do think that water = H2O is a metaphysical necessity. I’m not. Again, the issue isn’t what we would call such a substance, I really don’t care for semantics in this case. But I don’t see why you take the strong position that if something is made of hydrogen and oxygen it also must behave in a certain way, regardless of the laws of physics. Are you saying that for some reason we can alter, say, the strength of the laws characterizing molecular bonds, and yet somehow H2O would still behave in the same way? That sounds like magic to me… ;-)

> this question seems to me to be not metaphysical at all and it only depends on what you mean by water. I mean "H20" by water <

As I said, not interested in that angle. That is true, but trivially so, hence no further discussion is needed.

> I reject the idea of metaphysical necessity as distinct from logical necessity. <

I’m inclined to do the same, though I’m still somewhat open to the possibility of genuine metaphysical necessity. But it doesn’t sound from the above like this is your position.

> For something to be metaphysically necessary we would need a set of metaphysical laws. These laws could not be deduced from first principles because this would reduce them to logical necessity. They could not be detected by scientists because this would reduce them to physical laws. <

Yes, as I pointed out in the post.

> This is one of many reasons why I think all logically possible worlds must exist. <

Nah.

Thomas,

> “Finally, we will assume that whatever is metaphysically necessary is nomologically necessary, but not conversely.” <

Sounds like the author of the SEP entry distinguishes between physical (nomic) and metaphysical possibility, regarding the latter as ampler than the former. But on what grounds?

> I guess what I'm asking is whether an empirical approach might reveal a nomological necessity that is mistaken for a metaphysical necessity <

Well, since nomic necessity is a necessity by physical law, all nomic necessities are physical as well as metaphysical (just like if something is physically necessary it is also logically so). The question is whether there is something that is not physically, but metaphysically, necessary. Still looking for a good example…

>Then you do think that water = H2O is a metaphysical necessity.<No. This is only a semantic thing. For me, "water" means "H20". If you define "water" as a colourless, odorless liquid which expands slightly when it freezes, then I agree it doesn't have to be H2O, but that's not *my* particular definition.

>But I don’t see why you take the strong position that if something is made of hydrogen and oxygen it also must behave in a certain way, regardless of the laws of physics.<

That mischaracterises my position.I was suggesting that if it does not behave in that way because the laws of physics are different, then arguably it's not really made of hydrogen and oxygen, and that pseudo-hydrogen and pseudo-oxygen is not really made of electrons, protons and neutrons but their analogues in another world.

My actual position is that the question is meaningless, because it depends only on how you want to define "water", "oxygen", "hydrogen", "proton", "neutron", "electron", etc. Those all have well defined meanings in this universe, but if you're expanding the concept to include universes with other laws of physics those meanings break down and new definitions need to beagreed.

>Are you saying that for some reason we can alter, say, the strength of the laws characterizing molecular bonds, and yet somehow H2O would still behave in the same way?<I am absolutely not saying that. I am making a half-hearted case that it might not be H2O any more because it wouldn't be real hydrogen and oxygen if the laws of physics were changed. But if it's just tweaking a force that would be an unusual position to take. So assuming I accept that it's still hydrogen and oxygen, and that it's still H2O, I would say that it is still water even if its properties were very different. It would just be a new phase of water, like ice, liquid water and steam.

>As I said, not interested in that angle. That is true, but trivially so, hence no further discussion is needed.<

I think that is the only sensible angle for this question, which is why I don't think it's a very interesting question.

>> This is one of many reasons why I think all logically possible worlds must exist. <Nah.<

If you think that only one of the possible worlds exists, or that only some possible worlds exist, to me that implies that you believe in metaphysical law.

As far as I can see, rejecting metaphysical law means that all possible worlds must exist, because there can be no metaphysical level which defines which worlds exist and which worlds do not. Without metaphysical law, either all worlds must exist or no worlds must exist. Indeed the concept of physical existence as applied to worlds is revealed as incoherent, and the existence of worlds is simply equivalent to that which is logically consistent.

> As I pointed out before, Hume's criterion would have committed all his own writing to the flames. <

Oh, that’s the sort of cheap shot that people think doomed logical positivism (vis-a-vis its verifiability criterion). It doesn’t work because one can still do valuable work of conceptual criticism (i.e., philosophy) as a type of meta-work on the way humans think — as both Hume and the positivists were doing. (Full disclosure: I am NOT a positivist!)

No, Latin for “I shouldn’t need to explain this to you because it’s obvious.”

> Now a universe in which the structure and properties of atoms (which would include the structure and properties of all sub atomic identities) is identical to those in our own world but the macro behaviour is different? I am pretty sure that is prima facie ridiculous. <

Sarcasm aside, you didn’t read what I wrote carefully: I imagined a universe where the molecular structure of water is the same as in our universe, but the laws of physics regulating molecular interactions are different. I don’t see what’s ridiculous about that.

> The problem with what you are saying is that it implies that there is some necessarily true statement of pure logic that rules out the existence of things that are necessary but not logically necessary. <

Uh? I don’t think that’s what I said, at all. First of all, I’m not sure what’s the difference between pure and impure logic. Second, I said that if something is logically necessary then it better be both metaphysically (whatever that means) and physically necessary, though usually the statement is framed in the negative, about physical / metaphysical / logical impossibilities.

SelfAware,

> I'm almost certainly missing something here, but it seems to me that the only metaphysical *necessities* are tautological ones <

Maybe, but then there would be no distinction between logic and metaphysics.

> Physical necessities imply, to me at least, that we're no longer talking about metaphysics, but science of some sort. <

Massimo wrote: "No, Latin for “I shouldn’t need to explain this to you because it’s obvious.”

Not obvious to me. And, despite it's alleged obviousness nobody seems be able to come up with the explanation behind that obviousness.

Massimo wrote: "Oh, that’s the sort of cheap shot that people think doomed logical positivism (vis-a-vis its verifiability criterion). It doesn’t work because one can still do valuable work of conceptual criticism (i.e., philosophy) as a type of meta-work on the way humans think — as both Hume and the positivists were doing."

How is it a cheap shot to point out an inconsistency? I have to point out that you can do no valuable work at all with writings that have been committed to the flames as Hume's statement suggests.

If such work has value then Hume's statement is clearly wrong.

Massimo wrote: "Second, I said that if something is logically necessary then it better be both metaphysically (whatever that means) and physically necessary

I didn't get that from what you said, so you have thrown me a curve ball. The above statement appears to suggest that logical necessity implies physical necessity.

That neither appears to be what your post says, nor does it seem to be true. There seems to be a good deal of mathematical theorems which are logically necessary and have no physical application.

I thought that you were saying that the only metaphysically necessary things were also physically necessary (whatever that means).

I was basically saying that if this was true then (by its own thesis) it was either:

1. Not necessarily true2. Logically necessary3. Physically necessary

If it was logically necessary then it would imply some sort of logic that demonstrated its truth.

I am not sure what would be required for it to be physically necessary.

Oh, and to see the difference between pure and impure logic, think of Kleene's distinction between the object language and the observer language. Pure logic would have no statement in the observer language, just the object language.

Massimo wrote: "Sarcasm aside, you didn’t read what I wrote carefully: I imagined a universe where the molecular structure of water is the same as in our universe, but the laws of physics regulating molecular interactions are different. I don’t see what’s ridiculous about that.

I cannot see how you are disaggregating the description of a molecule with the laws of physics, that makes no sense to me.

Essentially you are saying that there might be something with some of the properties of H20 but not others which might meaningfully be described as being the same as H20.

You have got to understand Kripke's point - the identity between H20 and water is such that you would have to alter the description of one (just as Massimo is doing) in order to say that it is not water.

He was comparing this with the neurological underpinnings of pain. You could (if Naturalism is true) describe with perfect consistency a world in which all of our molecules work as they do in this world according to exactly the same physical laws and and producing exactly the same external behaviour and yet there was no pain in that world, and that this would involve no inconsistency.

So saying "c-fibres firing is not pain" is a perfectly consistent thing to say whereas "H20 is not water" is not.

(Of course the exact details of the neurology do not alter the point, it need not be " c-fibres firing" exactly)

>I imagined a universe where the molecular structure of water is the same as in our universe, but the laws of physics regulating molecular interactions are different. I don’t see what’s ridiculous about that.<

Thats the part I'm having trouble conceptualizing so I think I agree with DM and Robin. If the molecular structure of water in this possible world contains the same hydrogen and oxygen as in our world, that seems to imply the standard model would with all it's forces and particles would have to also be the same in the possible world. Does not the emergence of hydrogen and oxygen depend upon all of this. If the particles and forces operated on the same physics wouldn't the molecular interactions also have to be the same, or if some new or altered force affected the molecular level from where could it have emerged?

Massimo: "Well, since nomic necessity is a necessity by physical law, all nomic necessities are physical as well as metaphysical (just like if something is physically necessary it is also logically so). The question is whether there is something that is not physically, but metaphysically, necessary. Still looking for a good example."

I don't have a good example, and I am not in the least concerned that I don't. I'm thinking perhaps the problem is not with a modifier such as "metaphysical" or "physical," but perhaps the notion of "necessary" itself. Didn't Hume think of physical (natural) laws as metaphisically contingent rather than necessary while at the same time allowing for the possibility of different ones? So I'm suggesting that your statement, "our newfound way of thinking about metaphysical necessity either expands into logical necessity or collapses into physical necessity," at some future time may simply entail "bumping" or "expanding" terminology to accommodate refined or additional knowledge or information. In fact, we have already expanded our conceptions of both logic and what's physical. And so, it is conceivable that our current conceptualizations of what the terms logic and physical currently entail will be radically revised in the future. Will that radical revision result in additional modifiers for currently used terms or new adaptations of old terms? Will this lead us to make use of the term metaphysical necessity at some future time? I have no idea. In other words, I'm not sure you're not simply talking about explanatory power and what terms we currently use to convey it.

Without logical law, all statements are true and all statements are false.Without physical law, all future states are possible and all future states are impossible.Without metaphysical law, all possible worlds exist and no possible worlds exist.

Massimo, isn't: "That’s because metaphysicians these days seem to make sense of the notion of metaphysical necessity by saying that something is metaphysically necessary if it is true in all possible worlds" just a rewrite of logical necessity?

Not so. If there is such a thing as metaphysical law, then not all logically possible worlds exist. Possible worlds are those worlds which are metaphysically possible, a subset of the logically possible worlds.

It could be imagined, for instance, that God is a metaphysical necessity while not a logical necessity. It could just so happen that metaphysical law requires that there be God in all possible worlds. God is therefore metaphysically necessary without being logically necessary by the sentence you quote.

DM, supposing that there was an entity about which most of humanity was in agreement and called it God, would she have some say-so in what humanity described as metaphysically or logically necessary? Is this topic any clearer by introducing God into it? What you really mean is better served by "conceivable." What I mean to say is given a conventional meaning of "unicorn" and a conventional definition of God (if such definition could be possibly agreed upon), which conception would you prefer to support or refute as conceivable? If I suggested that you are favoring logic as opposed to empirical evidence to support your distinction between metaphysical and logical, your logic would nevertheless avail itself of what can be empirically proven. That in itself is a problem (not a significant one to me) because each viewpoint implicitly employs the other. The problem is in unpacking necessity and contingency and possibility and impossibility without a universally satisfactory conception of any of them. We have working definitions and laws, but sense that they inadequately correspond/cohere/represent what we describe as subjective experience.

Some have suggested that a proof for the Riemann Hypothesis might not even exist.

If true then the Riemann Hypothesis would be necessarily true.

But if it were necessarily true and there was no possible proof for it - not even the possibility of the statement that it is false involving a contradiction - then does that count as logically necessary?

>But if it were necessarily true and there was no possible proof for it - not even the possibility of the statement that it is false involving a contradiction - then does that count as logically necessary?<

Massimo wrote: "Still, there may be things that are metaphysically necessary in all possible instantiations of the multiverse. Perhaps the inter-conversion between matter and energy? Or the existence of fields from which matter emerges (like the Higgs)?"

The problem here is that if the "multiverse" in this case is the set of mathematically realisable universes then of course none of these things are necessary (true in all worlds) since there is no problem at all in realising a universe in which there is no matter, nor energy nor quantum fields nor any of the observed or conjectured aspects of our universe.

Mathematically you could realise a universe with nothing at all in common with this universe or multiverse (if there is one), except actual mathematical facts.

And so "physical necessity" just collapses into logical necessity.

So the problem is not to rescue something from Scylla or Charybdis, rather it is to rescue Charybdis from Scylla which it seems can't be done.

Now I am wondering - of the set of all mathematically realisable universes, what percentage would be ones in which there could be, even in principle, self organisation into complex structures. A vanishingly small percentage I am guessing.

If so and if the only necessity is logical necessity then it is not plausible that we exist at all.

So we need some kind of necessity other than logical necessity to explain us and physical necessity clearly will not do.

please see just added Postscript to the original post, it should stimulate some additional discussion!

Patrick,

> In the end there's a metaphysical necessity. But that arrives usually far past the moment when it became doubtful that all these Causes and Waters were describing anything except their own life in the metaphysical universe. <

I submit that that was just an example of bad metaphysics. For good ones, check the edited book by Ladyman & Ross linked from the post.

Seth,

> If the molecular structure of water in this possible world contains the same hydrogen and oxygen as in our world, that seems to imply the standard model would with all it's forces and particles would have to also be the same in the possible world. Does not the emergence of hydrogen and oxygen depend upon all of this. <

Not necessarily. Obviously, if you tinker with the laws of physics enough than you won’t have hydrogen and oxygen, or any heavy element, and so forth. But it is a rather uncontroversial assumptions of discussions about fine tuning that it is perfectly fine to talk about a universe with partially altered laws of physics which nonetheless would retain some of the characteristics of ours.

DM,

> This is only a semantic thing. For me, "water" means “H20". <

As I said, I’m not interested in semantics in this case. See my response to Seth above.

> I was suggesting that if it does not behave in that way because the laws of physics are different, then arguably it's not really made of hydrogen and oxygen <

Again, see above. If the laws of physics are not altered too much you simply cannot reach that conclusion.

> My actual position is that the question is meaningless, because it depends only on how you want to define "water", "oxygen", "hydrogen", "proton", "neutron", “electron" <

But it doesn’t, so the question isn’t meaningless at all.

> I accept that it's still hydrogen and oxygen, and that it's still H2O, I would say that it is still water even if its properties were very different. It would just be a new phase of water, like ice, liquid water and steam <

That’s better, I think. If that’s right, then water = H2O would indeed count as a metaphysical necessity, though in the specific sense that it is a *physical* necessity that applies to the multiverse.

> If you think that only one of the possible worlds exists, or that only some possible worlds exist, to me that implies that you believe in metaphysical law. <

Not at all. You neglect the possibility of stochasticity: some worlds happen to exist, others don’t.

> Indeed the concept of physical existence as applied to worlds is revealed as incoherent, and the existence of worlds is simply equivalent to that which is logically consistent. <

As others have pointed out, it’s not clear at all what you mean by “existence” here. Do you think all possible worlds exist physically, somewhere? Or are you essentially a modal realist? To me existence of a world means that it is somewhere, sometime, not just a logical abstraction.

Thomas,

> perhaps the notion of "necessary" itself <

I agree, I don’t think there is *anything* that is necessary, physical, metaphysical or even logical. There are, however, plenty of things that are physically and logically impossible (though I’m not sure about metaphysical impossibility, as a notion distinct from the other two).

> it is conceivable that our current conceptualizations of what the terms logic and physical currently entail will be radically revised in the future <

>> My actual position is that the question is meaningless, because it depends only on how you want to define "water", "oxygen", "hydrogen", "proton", "neutron", “electron" <

But it doesn’t, so the question isn’t meaningless at all.<

I just want to be clear on this. I am making a strong statement - the question of whether water is H2O is strictly a semantic question and it has no other meaningful interpretation. It is certainly not a metaphysical question, and I don't understand how it can be construed as such.

If you define "water" according to the emergent properties we associate with water, then H2O is not necessarily water. If you define water as H2O (as I do), then H2O is necessarily water. Either answer is reasonable, depending on semantics. I don't see any other way to interpret the question.

> You neglect the possibility of stochasticity: some worlds happen to exist, others don’t.<

Yes, that is a good point and I could see how you could make that argument, but I don't buy it. Even stochasticity has laws. What is the proportion of possible universes that exist (i.e. what is the probability that an arbitrary universe exists). How are they distributed? Do certain characteristics of universes make them more likely to exist than others?

There are also other problems with the argument. I'm not sure I buy that true stochasticity is really possible or coherent. Sure, you can point to quantum mechanics, but some interpretations of QM are fundamentally deterministic, and my preferred interpretation (many worlds) is among them.

But there's also the idea that each logically possible world is already well defined as an abstract object. What is it that confers this extra non-mathematical attribute of "existence" upon it if not some kind of metaphysical description of reality? Without allowing something fundamentally metaphysical into your ontology (metaphysical law or a metaphysical stochastic process of some kind) then I don't see how you can account for the existence of some possible worlds but not others.

>Do you think all possible worlds exist physically, somewhere? Or are you essentially a modal realist?<

Yes to both! These are not mutually exclusive. Indeed, as I understand it, the latter alternative is just a label for the former.

I know you have a lot of regular commenters on this blog, but I'm somewhat disappointed that you don't know by now that I adhere to the Mathematical Universe Hypothesis, which can be construed as a kind of robust and well-defined form of Modal Realism, where the logically possible worlds are simply the well-defined mathematical objects.

> isn't: "That’s because metaphysicians these days seem to make sense of the notion of metaphysical necessity by saying that something is metaphysically necessary if it is true in all possible worlds" just a rewrite of logical necessity? <

Either that or we are left with no metrics to compare possible worlds. Which, in fact, is a big problem in modal logic and possible worlds semantic.

Philip,

> We have languages for writing logical and physical laws, but what is a language for writing metaphysical laws? If none, then there are no metaphysical laws. <

Indeed, and I incline toward that answer myself.

Robin,

> Not obvious to me. <

I’m sorry, but the burden of proof is on you: I can’t even make sense of the very idea that a god (whatever that is) is logically necessary. Until then, it is nonsense on stilts to me.

> "Necessarily existing" is part of the description of the God posited by the major monotheistic faiths. <

Thanks, Massimo. I wanted to double-check that I wasn't hallucinating, misreading, or missing something. I appreciate DM's response to me, but think it falls short. Agreed with the additional insight on modal logic, which in turn is another reason why I think conservative Xns who blather about the brilliance of Plantinga are overrating him.

Apologies for the confusion, as I wrote above, I should actually have phrased it in terms of impossibility, not necessity. I don’t think there is anything that is necessary, logically or otherwise.

> to see the difference between pure and impure logic, think of Kleene's distinction between the object language and the observer language. Pure logic would have no statement in the observer language, just the object language. <

No, sorry, I still don’t see it. Logic is logic, the term “pure” is irrelevant.

> I cannot see how you are disaggregating the description of a molecule with the laws of physics, that makes no sense to me. <

See my response to Seth above.

> I am not sure what "physical necessity" could mean unless it were "physical necessity given that this particular set of physical laws are the case”. <

Indeed. And I don’t see any reason to think that this particular set of physical laws was necessary, so there is no such thing as (meta?) physical necessity.

> The problem here is that if the "multiverse" in this case is the set of mathematically realisable universes then of course none of these things are necessary <

It isn’t, it’s a subset of that mathematically realizable universe.

> there is no problem at all in realising a universe in which there is no matter, nor energy nor quantum fields <

Not much of a universe, eh?

> So the problem is not to rescue something from Scylla or Charybdis, rather it is to rescue Charybdis from Scylla which it seems can't be done <

Not if you think in terms of impossibilities, rather than necessity, as I suggested above (again, sorry for the earlier confusion generated by my hastiness).

> If so and if the only necessity is logical necessity then it is not plausible that we exist at all. So we need some kind of necessity other than logical necessity to explain us and physical necessity clearly will not do. <

I just wanted to say that I very much enjoyed and agreed with your addendum to the main post. In particular, I am very much with you on the role of metaphysics, which I think you expressed beautifully. I agree with you that it is an important role, and in fact it is my main interest in philosophy.

However, while I agree that there is no one correct set of axioms, I think there is still a sense in which one can say that the laws of logic are necessary. When I say this, I mean that as soon as you identify your axioms, what you conclude from that is necessarily constrained. I do not accept for example the idea that there are possible worlds where one can reach different conclusions from the same axioms. I find it necessary to say this sometimes because I have encountered people who disagree.

I agree with the last part of what you say. There is no.possible circumstance. no possible version of reality in which there is an algorithm to enumerate the digits of a chaitin constant. In that sense there is logical necessity.

Massimo, thanks for the helpful postscript. I think I now understand better the framework of your post.I am particularly drawn to this: "To begin with, I am leaning toward the conclusion that there is no such thing as metaphysical necessity. That’s in part because one cannot find metaphysical laws, and in part because I doubt there is such a thing as necessity, period. Nothing is physically or logically necessary - only possible or impossible."

For better or worse, that seems to describe the predicament in which I often find myself these days.

Massimo, I don't recall ever saying that we are necessary and so I am puzzled as to why you attribute that to me.

But we are here, as you say.

And if I understand you correctly now you are saying that physical reality is the way it is - even at the most fundamental level - for no reason at all.

Which makes me wonder again - of all possible mathematically realisable realities what percentage would there be the possobility, even in principle, for self organisation to higher levels of complexity.

And the idea of "observer self selection" does not apply here. If the actual world was one of the possible worlds where self organisation to higher levels of complexity was not even in principle possible then there would be no observer to self select.

How so? Proof of what? What claim did I make? Is a proposition ludicrous until proven otherwise?

Massimo wrote: "Not much of a universe, eh?"

How do you know? It might be a very magnificent universe indeed.

Massimo wrote: I explained it

No you didn't. If you are agreeing with Hume's criteria and also saying that books which don't meet those criteria have value then you are saying that books which have value ought to be committed to the flames.

Logical necessity consists of the fact that certain things follow from certain axioms. I cannot see how this is altered by the fact of those axioms having been chosen or not chosen.

So we know there are logically necessary truths, for example the square of the length of the hypotenuse of a right angled triangle defined on a cross product of reals is equal to the sum of the square of the lengths of the other two sides in every possible world.

There is no possible world in which there is an algorithm which enumerates the digits of a Chaitin constant.

Metaphysical necessity would consist of there being a fact that was true in all possible worlds and which was not logically necessary.

If we do not know of such a fact we cannot infer from this that there is no metaphysically necessary fact, only that we do not know if there is one.

I would suggest that there must be because there are logically possible worlds in which self organisation to greater complexity are not even in principle possible.

I believe I can demonstrate that such worlds would vastly outnumber worlds in which self organisation is in principle possible.

If there were no fact of the matter as to why our reality is the way it is, beyond logical necessity, then the fact of our living in a reality in which self-organisation to greater complexity is possible would be vastly improbable.

But we are living in such a reality and so I would tend to favour the proposition that there is at least one thing that is metaphysically necessary.

This is not an argument for the existence of God because the only things I can infer about this thing is that it is metaphysically necessary and it mandates that self-organisation to greater complexity is possible.

The section you added at the end was also helpful in delineating where you stand as far as the idea of necessity goes. One thing I don't quite understand with that position is that there do seem to be some logical statements that really are 'necessary,' even though some might consider them tautologies with no real value (which, as you rightfully point out, is totally off the mark). For instance, the equation x = x or some other form of 'identity relation' is something that I think one can make a very strong case for being logically 'necessary.' I mean, what the hell would it even mean for that to be a possibility amongst worlds? I guess if one takes a somewhat formalist view of mathematics/logic then you can banish necessity and say we're playing with different axiomatic systems.

I don't really share that view, and I know that many logicians and mathematicians working on foundational questions believe that there is one all encompassing world of mathematics/one true logic. Regardless of whether or not that is even true, I would say there are at least a few very very intuitive notions like the identity relation that would be true in every single world imaginable. Wouldn't those then be necessary laws of existence? Isn't the whole idea of logical truth (at least with some of the most primitive tautologies) that these truths hold in every single logically possible world, not simply that they're 'possible' in every possible world?

I might be totally off on my interpretation Massimo, but I'd welcome some input from you or any others.

I certainly don't think there is one true logic. I expect there are axiomatic systems that don't include the identify relation.

But these systems do not contradict more familiar logics, they sit alongside them. Where we define our axioms differently so that familiar symbols such as AND, OR, NOT, = take on slightly different semantics, we are simply not talking about the same logical operators. The reuse of the familiar symbols makes it look like we are contradicting ourselves but instead we're just talking a different language, and both axiomatic systems can exist side by side with both being valid at the same time.

Given a specific set of axioms, the logic that follows will indeed be the same in all possible worlds. What Massimo means by denying logical necessity is that there is no one true set of axioms. All consistent sets of axioms are equally valid, no matter how strange or unintuitive they first appear.

Yes, all we can really say about logic is that it means that we select a collection of symbols and that we invent some rules by which to manipulate them. The difference between logic and mathematics generally is, I suppose, it's discrete nature but even that only has a meaning consequent upon some selection of symbols and rules.

Also we could replace "true,false" with "Heckle,Jeckle" (for three valued logic we might use "Huey,Duey,Louis") so that instead of the Pythagorean Theorem (or whatever consequence of whatever symbol manipulation we choose) being "true", it is instead "Heckle".

But none of that contradicts the idea that there is logical necessity, it only unpacks the concept.

Also, I should point out that the "Heckle/Jekyll" thing does not mean that there is no logical truth, only that logical truth means something in the observer language and not necessarily in the object language.

I'm a bit surprised that you wouldn't be a proponent of the 'one true logic' idea, as I know from other threads on mathematics/logic/metaphysics that you're definitely a mathematical Platonist. It's a bit weird to hold that worldview while also maintaining the type of formalism that I would assume you're suggesting. If what you say is true then the continuum hypothesis is true in some axiomatic systems of mathematics and false in other systems, but then how could you ever say that CH is "true" in the sense of being an objective fact of the mathematical (dare I even suggest physical) universe?

This is a big question in set theory and the foundations of mathematics, and it has been for decades. Godel, and many other platonists as well, would say that there is a fixed body of infinitely many mathematical truths that exist. Godel even suggests that though we can never know all of them completely (a product of his own theorems) we can over time add more and more axioms to include more and more mathematical truth. This would allow us to constantly illuminate the world of objective mathematical truth, though we could obviously never uncover everything.

I tend to subscribe to that view, as would many other mathematicians working in set theory today. Hugh Woodin has even made some breakthrough progress with the idea of Ultimate L, which I've mentioned in previous posts I believe. I've read as much as I can about this idea (all non-technical of course, as set theory is hard as shit) and it looks very enticing to many in the field. It would basically be an all encompassing picture of the universe of sets, giving a full account of what they are and include within it any possible construction. I should mention that it would still be incomplete, as other axioms would need to be added to prove more, but the 'universe of sets' would be totally incorporated within the framework.

There are some set theorists who are Platonists while still maintaining a multiverse view of set theory, with some universes having the CH be true while others do not. This looks to be the minority position, and in all honesty I really cant understand how the mathematical universe could be partitioned into "CH is true over here" and "Oh yea but its false over there!"Before you mention the discovery of Non-Euclidean Geometry and try to go down that route remember that all of that is still included within the ZFC framework, so its not really different mathematical systems at the most fundamental level of sets. I just can't buy the idea that random assortments of axioms produce equally true mathematical statements. Here are some links that might be of help:

>It's a bit weird to hold that worldview while also maintaining the type of formalism that I would assume you're suggesting.<

It's not really formalism, as I believe that all formal systems and all mathematical objects definable with those systems actually exist (which is anti-formalism). It's known as plenitudinous or full-blooded Platonism.

>but then how could you ever say that CH is "true" in the sense of being an objective fact of the mathematical (dare I even suggest physical) universe? <You couldn't. You say it is true in this system and not in that. Both systems exist. It's like the idea that there are different laws of physics in different universes, both of which could exist.

As I mentioned before (Re: Tegmark), there appears to be a lot of activity now in using the language of types (over sets), unifying constructive logic, programming, and category theory. This is a different way of expressing mathematics.

I can definitely understand where you're coming from in suggesting FBP and trying to tie it back into the example of different universes with different laws of physics. The problem with this is that it doesn't actually work. See, in the idea of a multiverse where different physical laws are instantiated in different universes (think of the 10^500 possible Calabi-Yau configurations or whatnot), there is still actually a meta-theory or meta-laws of physics. You can't escape that. Whether its M-theory of some other type of overarching framework, the different universes with different laws are all still governed by the meta-law, and they evolve in all different sorts of ways according to that meta-law.

This same type of argument applies to mathematics. There should be an overarching framework that unifies and provides the proper foundation for deciding objective mathematical statements. It makes no sense to think that over here a particular mathematical statement is true, but over there its not (in the abstract mathematical universe). I mean at least in the case of multiverses there are physical pockets with different underlying geometrical entities that give rises to different particles/forces. Where would this demarcation exist in the abstract realm of mathematics? I find it hard to believe there is one.

Phil -

Good point with the mentioning type theory/category theory. The foundations of mathematics are definitely still being developed, and who knows what method will garner the most fundamental and unifying picture of the mathematical universe.

Robin -

You have to put the idea of it all being "symbol manipulation" to rest. It's very clear that that's actually not what it is at all. These are real structures that are inherent in the physical world. Hell we can make predictions from pure math, and those patterns/forces/particles are only discovered later. What gives these symbol manipulations that type of power? Sure we describe them with symbols and that's what we use to communicate with each other, but you have to look past that.

Pete, I can put the idea of it all being a symbol manipulation to rest just as soon as someone gives me a good reason for doing so.

What gives the symbol manipulation it's predictive power? Nobody knows, because you cannot get past the symbol manipulation.

We may refer speculatively to the "thing in itself", the "somewhat" the "physical world" as it has variously been called, but the most information we have about it, probably that we will ever have about it, is a symbol manipulation.

There is no logical requirement for there to be something that the symbol manipulations are referring to. There is no need for that hypothesis in science. In fact some of our most important science was done by physicists who made no assumption about this.

We could not in any account know anything about the "what the symbol manipulation describes" beyond the symbol manipulation itself and the sensations we have of it.

"I can definitely understand where you're coming from in suggesting FBP and trying to tie it back into the example of different universes with different laws of physics. The problem with this is that it doesn't actually work. See, in the idea of a multiverse where different physical laws are instantiated in different universes (think of the 10^500 possible Calabi-Yau configurations or whatnot), there is still actually a meta-theory or meta-laws of physics."

I think you may be mistaking my analogy to physics for an argument when it is intended only as a means of illustrating my position. Yes, there are differences between the set of formal systems and the set of universes within a specific kind of multiverse. That doesn't necessarily kill the analogy though. All analogies involve aspects of similarity and aspects of difference. The question is whether the differences are more relevant than the similarities, and I don't think they are, at least not if I'm only trying to explain my meaning.

Besides, there is an underlying meta-theory for the set of all formal systems, and that is the rule that mathematical objects described by consistent formal systems exist.

A core idea of the MUH is to discard all unnecessary axioms and metaphysical laws. The one law that remains is arguably not a law at all but a principle with respect to how we regard the concept of existence. In the multiverse of the MUH, therefore, your criticism does not apply.

"It makes no sense to think that over here a particular mathematical statement is true, but over there its not (in the abstract mathematical universe)"

I agree, which is why I think that there are different interpretations of my position at play.

I think of the different formal systems not in terms of being different places but more like being different languages. It makes no sense to imagine that the correct word for "Hello" is "Hola" in Spain and "Bonjour" in France. But it does make sense to say that the correct word is "Hola" in the context of Spanish, and "Bonjour" in the context of French.

So the idea is that there is no absolute mathematical truth without context. Every mathematical utterance is made in the context of some formal system whether implicit or explicit, and its truth is evaluated accordingly.

I think that you need to unpack what sort of thing an axiom or a formal system would be in the mathematical universe.

In terms of the mathematics I do on paper these are sets of rules in a language and depend upon an intelligent mind to interpret them and agree on meanings.

Unless you are positing a default mathematician, which I know you are not, then you need to be clearer about what is the analog of these things you are proposing in the self existing mathematical realm.

An axiom is part of the definition of a mathematical object. If you view an axiom as a statement for human minds to appreciate and interpret, you are thinking about the representation in symbols of a mathematical concept and not about the mathematical concept itself. The axiom is the meaning agreed upon by those who use it, not the statement used to represent it.

Thanks for clarifying. I must say that the language analogy is a bid misplaced as well. Languages are used for communicating and there are many different ways of doing this. However, as far as being descriptive of certain aspects of the world around us, they still refer to objective features of existence that are unique and constant. Mathematical truth shouldn't have anything to do with the context of human beings communicating with each other.

Again, it's an analogy to try to communicate something about how I think of it, not an argument. It's not the same thing as language, but I think it is in some respects more appropriate to think of it like a language than a place. My view of this does not in fact depend on human communications at all.

One interesting thing I think languages and formal systems share is that just like languages, different formal systems can be used to describe the same mathematical objects. ZFC and ZF can both be used to describe the Mandelbrot Set even though these are different formal systems, and in my view there is only one Mandelbrot Set.

Even better than a language would be to think of it as a context. The distance between geodesics remains constant in the context of Euclidian geometry, but not in the context of Hyperbolic geometry, but there is no conflict because we are talking about different contexts. This is perhaps reasonably easy to understand. You seem to think that the Continuum Hypothesis is fundamentally different, but I will only accept that if its affirmation or its negation turns out to be logically incoherent. The CH is true in the context of ZFC+CH and false in ZFC+¬CH, just as the parallel postulate is true in Euclidean and false in Hyperbolic geometry.

>"That’s in part because one cannot find metaphysical laws, and in part because I doubt there is such a thing as necessity, period. Nothing is physically or logically necessary - only possible or impossible."<

Hmmm. As I see it, logical necessity is grounded in contradiction: roughly, if p entails a contradiction (without contingent premises) then ~p is logically necessary.

As to metaphysics, laws of metaphysics are not needed for metaphysical necessity. (The notion of a metaphysical law may not make sense.) All that is needed is something that could rightly be called a metaphysical truth. If p contradicted such a truth, ~p would be metaphysically necessary. I am not sure however that there are truths that can rightly be called metaphysical; on that point we may agree, and I agree about the matter of such a truth falling into either logical of physical necessity.

As to whether or not there is such a thing as necessity, as necessity is obviously not a thing, it might be helpful to clarify exactly what the concern is here. The concern seems to have two parts: first, does the predicate 'necessary' have a coherent meaning?; second, is the predicate true of anything? Maybe what you mean by there being no physical necessity is that this predicate has no coherent meaning; or perhaps you believe it has a coherent meaning but do not think it is true of anything.

I would say that there are physically necessary propositions if there are propositions that contradicts the laws of physics; i.e., if p contradicts the laws of physics then ~p is physically necessary.

So it seems to me that you're claiming that there are no propositions that contradict the laws of physics.

But what about this: If it is physically impossible for any x to travel faster than the speed of light, then it is physically necessary that for all x, x travels at most at the speed of light.

(This also suggests how necessity and impossibility are logically interlinked.)

>"The same goes with logical necessity: once we pick certain axioms or premises, a number of things necessarily follow. But we could have picked different axioms or premises, so that those very same things wouldn’t follow at all."<

On the contrary. First, as a more general point, the matter of logical necessity doesn't have much to do essentially with logical systems. Logical systems are not logic itself but simply symbolic systems that are themselves subject to the reality of logic, as natural languages are. This reality falls out of the nature of language, whether formal or natural - the phenomenon of semantic contradiction (i.e. incompatible meanings), in particular, I think. Entailment, and other things, such as the modalities, are at root about contradiction.

Btw, if there were no logical necessity, there would be no difference between material implication and entailment, which would mean the notion of a valid argument is empty; there would be just "contingent" if-then statements. I put 'contingent' in quotes because it would lose sense too.

As a more particular criticism of your above statement, within logical systems, logical truth, a realm of logical necessity, is a matter of semantics rather than deducibility; what is or isn't deducible within a system is independent of which formulas are logical truths. This distinction was critical for Goedel's completeness theorem for first order logic in that it establish that a formula is a logical truth if and only if it is provable: i.e. |=P iff |-P. In any system of pure logic (i.e., no non-logical axioms), it would be hoped that all axioms and all formulas deducible from the axioms are logical truths. Choice of axioms in such cases isn't a matter of choosing what is logical necessary but of choosing which logical necessities to include. We can no more change logical necessity by changing axioms than we can change physical reality by changing the formulas of physics.

Robin, but if the negation of a proposition involved no contradiction its hard to see what could be meant by logically necessary. After all, if the negation involved no contradiction, couldn't we say it is at least logically possible, and hence that original proposition is not logically necessary?

This seems to be a matter of choice of definition as I discussed with Disagreeable Me above.

The Riemann Conjecture is either necessary or false, whether it is logically necessary or not depends upon whether to include contradiction in the definition. Disagreeable Me preferred a definition which did not require contradiction.

I think the following article in SEP will help some of the commentators better understand Massimo's post. And, in fairness to him, keep in mind his statement, "At least (again), this is what I think this week." After reading the article, I think one can appreciate the tentativeness of his statement.

I don't follow. Why would one use such a phrase? To distinguish between facts that are necessary as opposed to those that are unnecessary? To contrast those that are deemed significant with those deemed trivial? I don't see that this is helpful, Robin.

Robin: "If it is possibly false that there is nothing that is necessarily true then there is something that is necessarily true."

And I don't accept this. I'm not even certain that it makes sense. I have a suspicion that you want to say something that is declarative, but opt instead for indirection. At any rate, I cannot discuss what I don't understand, but am willing to accept greater responsibility for this failure.

This is the sort of thing that Wittgenstein might have had in mind when he wrote in PI: "101. We want to say that there can't be any vagueness in logic. The idea now absorbs us, that the ideal 'must' be found in reality. Meanwhile we do not as yet see "how" it occurs there, nor do we understand the nature of this "must". We think it must be in reality; for we think we already see it there."

I hate to interrupt the conversation midway but I would like to ask a question that relates tangentially to the topic at hand. I have spent the last couple of years trying to follow various philosophy centric blogs, podcasts and the like as I find many of the topics interesting but also to make some judgement of the utility of philosophy. I know how that sounds rather pretentious and possibly dismissive but I come from a scientific background and have to admit a strong bias towards utility. So my question is as follows. Even though it appears unlikely that a definite conclusion will be found and agreed to in this discussion let's assume that one is deduced. What would be the usefulness of this conclusion? By that I mean, how would this new piece of philosophical knowledge be used in any sphere of human endeavor? Again, I am just trying to understand the the importance of this discussion and how it fits into the larger philosophic mosaic and whether there is any application outside of philosophy whether directly on indirectly. Thanks and sorry for the interruption.

Logic is what underlies mathematics and mathematics is what underlies physics which has quite a good deal of utility.

There have been completely pointless sounding arguments which have proved to have utility. For example a twelfth century monk comes up with an ingenious argument for the existence of God.

It sounds like rubbish but no-one can quite say why it is rubbish and the eventual refutation of it led to the biggest development in logic since Aristotle.

Much of our mathematics could not be stated without this advancement in logic and mathematics is, as I said before the basis of physics which has quite a lot of utility.

I could make a good case that the whole movement of Logical Positivism had an influence on science in that physicists of the early 20th century focussed less on concepts of reality and more on whether models worked.

So I guess it is a little like science or mathematics. This discussion may have no utility at all, or it may have utility in ways which we are not expecting but all ideas, no matter how apparently unrealistic, can have utility as the Anselm example above shows.

For me it has an inherent value which can also be utility since the concept of utility also involves personal utility.

But I wonder why you link a background in science to a bias toward utility. Engineering maybe, but scientists are generally more likely to be moved by thinga like fascination, awe and the inherent value of knowledge than they are by utility.

But there are no laws of metaphysics, just like there are no laws of philosophy, so this endeavor is one of critically making sense of things, not of discovering or dictating how things are.

That would be correct, but who know whether you know why? The answer is you can only arrive at such speculations using rationality and they only persist as long as rationally satisfying to you in making sense of logic and physics. Rationality is a means, or ongoing process, with no end of laws in itself. Strictly speaking.

“…metaphysicians these days seem to make sense of the notion of metaphysical necessity by saying that something is metaphysically necessary if it is true in all possible worlds. … I am leaning toward the conclusion that there is no such thing as metaphysical necessity. That’s in part because one cannot find metaphysical laws, and in part because I doubt there is such a thing as necessity, period. Nothing is physically or logically necessary - only possible or impossible.”

You have given two different ‘definitions’ for ‘metaphysical necessity’ while they are internal consistent between them. Of course, I cannot disagree with you if I use these two definitions to discuss this issue. Thus, if you allow, I will make a new definition.

I will define this issue with two steps.First, finding out the concreteness.Two, finding out the ‘ultimate’ concreteness.It will take a 200 page book to talk about this concreteness. Thus, I will not go into the details of it in this short comment but just point out a few key points. The concreteness does not depend on the physics laws, math theory, reasoning, logic, time, eternal, or space. I am here now [will definitely be gone in a future time, that is, my being is just a possible outcome in Buddhism], and it is ‘concreteness’ for me and to anyone who interacts with me. A 126 Gev. particle were seen by 6,000 physicists at LHC, it is a concrete object for them. Now, we can make the ‘first’ metaphysical law (FML).

FML --- for every concreteness, there is a something as the ‘metaphysical necessity’ for that concreteness. Corollary --- if a concreteness is not the ‘ultimate’ concreteness, it cannot be its own metaphysical necessity.

Now, with a unlimited concreteness set (UC) = {C1, C2, … Cn,…}, C(u) is a standalone single concreteness, and there is a set (in finite numbers) of rules which allows the C(u) to give rise to all members of set (UC), then C(u) is the ‘ultimate’ concreteness.

Of course, there is a big issue about whether such an ultimate concreteness can be found. Yet, with these new definitions, we might be able to address this issue in a new direction.

Since you introduced Buddhism, can you elaborate on how one would arrive at ultimate concreteness from notions such as dependent co-arising or dependent origination or inter-dependent arising? I'm not being sarcastic. I'm just wondering how to reconcile this concreteness with Buddhist notions of impermanence and change.

“I'm not being sarcastic. I'm just wondering how to reconcile this concreteness with Buddhist notions of impermanence and change.”

Excellent question. In fact, my definition does not meet the requirement that the metaphysically necessary of ‘my concreteness’ is true in all possible worlds, at the first glance.

Let’s start with your concreteness. Before I know you, your concreteness has no ‘meaning’ to my concreteness, that is, I can deny your concreteness without violating any physical or moral laws. Yet, as soon as I click the ‘reply’ button, a new concreteness arose, the interaction between us. There are two issues about this new concreteness.First, your concreteness can no longer be denied by my concreteness.Second, there is a ‘metaphysical necessity’ for this newly arose concreteness, and it consists of at least the followings:a. There must be a space which separate us.b. There must be a thing called ‘individuality’ which separate us in addition to space.c. Massimo Pigliucci must be a concreteness which I no longer can deny.

Very quickly, this ‘local’ and transient concreteness is no longer local and transient anymore. Now, I will introduce the 2nd metaphysics law.

Law 2: If concreteness B is undeniable by concreteness A, then the ‘metaphysical necessity’ of concreteness B cannot be denied by concreteness A.

Very soon, this ‘metaphysical necessity’ of concreteness B cannot be denied by many, and it is not too difficult to show that it cannot be denied in all possible worlds if they are also concreteness.

Yet, I showed that there are two different types of concreteness.One, the “I”-type concreteness which can be denied in practice although not in principle because of the Law 2.Two, the ‘ultimate’ concreteness (the ‘u’-type) which by definition must give rise to all I-type concreteness.

Now, I must introduce the Law 3 --- The ‘metaphysical necessity’ of an I-concreteness must also be a concreteness.

By definition, there is a u-type concreteness. But, is there a u-concreteness? Let me start with a subset of set (UC), the set (4), and it consists of only four concreteness.C1 = Alpha fine structure constantC2 = (Dark/visible) mass ratioC3 = particle zoo of Standard ModelC4 = I am here (a “life”)

Theorem one: If concreteness X is the ‘metaphysical necessity’ of set (4), it is a u-concreteness. Is concreteness X a concreteness? Can we find it? Of course, we can.How about the “Buddhist notions of impermanence and change”? A good ancient ‘opinion’, it does describe one point of view.

Tienzen, thank you for taking the time to elaborate on the term concreteness. I have tried to substitute various other terms, such as instantiation or relationship, in an attempt to grasp it (concreteness), but without much success. I searched the internet for a usage other than its conventional use and can't find one. And so, I assume it has an idiosyncratic meaning that you have tried to establish and to explain. I think it may be that your discussion is simply beyond my ability to comprehend it.

Tienzen, my apologies. I searched SEP for articles that mention "concreteness," and found at least 32 articles in which it was mentioned in one context or another. I am clearly in over my head. I am sorry I cannot engage in a fruitful discussion with you regarding your comments.

@Thomas Jones:“I searched SEP for articles that mention "concreteness," and found at least 32 articles in which it was mentioned in one context or another. I am clearly in over my head. I am sorry I cannot engage in a fruitful discussion with you regarding your comments.”

As these were just short comments, I did not give out more precise description about what I mean on ‘concreteness’, although I did know that it can be described in many different contexts. Thus, I took an easy way out, by using an operational definition, the “I”-concreteness (similar to Descartes' "Cogito ergo sum"). I further hinted that the u-concreteness is the ‘ultimate reality’ without giving details.

In addition to the operational definition, ‘concreteness’ has special meanings to me. One, mathematically, it is a precise procedure of how an infinity concretizes into a finite ‘object’. Thus, a finitude (finite universe) can be risen from infinity. Please see http://prebabel.blogspot.com/2013/10/multiverse-bubbles-are-now-all-burst-by.html .

Two, with Cantor’s theorem, it is a precise procedure of how multi-dimensional universe arose from one-dimension. See, http://prebabel.blogspot.com/2012/04/origin-of-spatial-dimensions-and.html .

Three, for u-concreteness, it must have, at least, the following attributes.a. It must be eternal and immutable,b. It must give rise to all the followings: (the cosmological constant (Λ), the Cabibbo angle, the Weinberg angles, the Alpha, the Neff = 3, the Planck data, the particle zoo of the Standard Model, the Baryongenesis, the quantum principle, the unified force equation (including gravity), the life, the Quantum-Spin, Why is here something rather than nothing? And more.) See, http://www.quantumdiaries.org/2014/02/07/interpretations-of-quantum-mechanics/#comment-172443 .